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There are many push/fold charts that have been calculated which claim to be optimal in a simplified version of (no limit texas holdem) where only push/fold is possible.

So wouldn't always either folding or pushing with a certain range be an optimal strategy (in the sense of EV>=0 against any other strategy)? Because it would force your opponent(s) to play something very similar to a push/fold game.

Would this work in general? If not, would this still work for heads up no limit holdem? And if not, why would an optimal strategy in general need calling or checking or non-all in betting?

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  • To put it really simply, an optimal strategy is to take advantage of your opponent while making it as close to impossible to be taken advantage of ourselves. I'm not going to put a full answer up as I don't have a lot of time right now, but the reason why it may be optimal to call, check or just bet a not all-in amount is because you balance your range and it may be the better play to also take advantage of your opponents range. Only having push or fold moves is exploitable. What if someone shoves on you? They can call with a monster, wait for your shove, and happily call for max value. etc
    – Grinch91 ♦
    Aug 5, 2020 at 13:19
  • Push fold charts are valuable, especially in hyper SNGs, but don't take them in a vacuum. They can help you make better decisions for max equity when you are in push fold situations, but that doesn't mean every situation should be a push/fold situation, even if a chart says you should. I.E. player who has played 1 hand in the entire game all of a sudden opens and you have a hand that is like just in the push range. This one you should probably fold.
    – Grinch91 ♦
    Aug 5, 2020 at 13:21
  • @Grinch91 im asking in a purely theoretical sense, assuming oponent's strategies can be arbitrary (future actions might be independent from past actions, unlike what usually happens with human opponents). Even against a player who does that, the push/fold chart should have non-negative EV. i'm interested in knowing if there is a strategy with non-negative EV against any other strategy in which you never check call or non-allin bet .
    – Carla Cvekla
    Aug 5, 2020 at 15:37

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It's not optimal. For NL hold'em, consider when each player has a stack size several thousand times the big blind. If you're limited to push-fold, then here's my strategy:

  • In the big blind, I only call with aces, and fold with everything else.
  • On the button, I min-raise with everything. I then call with aces, and fold anything else.

You can only push with aces, anything else loses you money. So when you have aces, you win on average 1.5 BB. Anything else, and I win on average .75 BB. Obviously, this is hugely profitable for me.

(I'm assuming that when I'm on the button, if I call you can check and see a flop. If you literally can't check, but either push or muck, then I call on the button, and I only lose 1 BB on average when you have aces)

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